Official iOS firmware for iPod touch (2nd generation), served directly from Apple. 8 IPSW files are listed below, including developer betas and release candidates.
How to install iOS on iPod touch (2nd generation) with an IPSW file
Back up your iPod touch (2nd generation) first. Connect it to a computer, open Finder on macOS or iTunes on Windows, then hold Option (Mac) or Shift (Windows) while you click Restore and pick the IPSW file. Your device will be erased and the selected version installed. Follow our restore guide for every step.
Only firmware marked as Signed can be installed today. To leave a beta and return to the public release, restore the latest signed release build listed above.